Abstract

Intense beams of light propagating through a medium with a positive Kerr nonlinearity can undergo self-focusing provided that their average power is larger than a certain critical power determined by the wavelength and material properties of the medium. Here, we show that for pulses comprising only a few optical cycles, this self-focusing can be inhibited by the presence of significant (normal) dispersion. We derive simple expressions to quantify the threshold power for self-focusing in the presence of dispersion. In addition, we show that under certain conditions, this threshold power can be larger than conventional critical power (for a dispersionless case) by a factor as large as several hundred.

© 2019 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2019 (3)

A. N. Tcypkin, M. V. Melnik, M. O. Zhukova, I. O. Vorontsova, S. E. Putilin, S. A. Kozlov, and X.-C. Zhang, “High Kerr nonlinearity of water in THz spectral range,” Opt. Express 27, 10419–10425 (2019).
[Crossref]

M. Melnik, I. Vorontsova, S. Putilin, A. Tcypkin, and S. Kozlov, “Methodical inaccuracy of the z-scan method for few-cycle terahertz pulses,” Sci. Rep. 9, 9146 (2019).
[Crossref]

S. W. Jolly, N. H. Matlis, F. Ahr, V. Leroux, T. Eichner, A.-L. Calendron, H. Ishizuki, T. Taira, F. X. Kärtner, and A. R. Maier, “Spectral phase control of interfering chirped pulses for high-energy narrowband terahertz generation,” Nat. Commun. 10, 2591 (2019).
[Crossref]

2018 (3)

P. Nugraha, G. Krizsán, G. Polónyi, M. Mechler, J. Hebling, G. Tóth, and J. Fülöp, “Efficient semiconductor multicycle terahertz pulse source,” J. Phys. B 51, 094007 (2018).
[Crossref]

A. N. Tcypkin, S. E. Putilin, M. C. Kulya, M. V. Melnik, A. A. Drozdov, V. G. Bespalov, X.-C. Zhang, R. W. Boyd, and S. A. Kozlov, “Experimental estimate of the nonlinear refractive index of crystalline ZnSe in the terahertz spectral range,” Bull. Russ. Acad. Sci. Phys. 82(12), 1547–1549 (2018).
[Crossref]

C. L. Korpa, G. Tóth, and J. Hebling, “Mapping the lattice-vibration potential using terahertz pulses,” J. Phys. B 51, 035403 (2018).
[Crossref]

2017 (1)

T. Seifert, S. Jaiswal, M. Sajadi, G. Jakob, S. Winnerl, M. Wolf, M. Klaui, and T. Kampfrath, “Ultrabroadband single-cycle terahertz pulses with peak fields of 300  kv  cm-1 from a metallic spintronic emitter,” Appl. Phys. Lett. 110, 252402 (2017).
[Crossref]

2016 (2)

M. Shalaby, C. Vicario, K. Thirupugalmani, S. Brahadeeswaran, and C. P. Hauri, “Intense THz source based on BNA organic crystal pumped at Ti:sapphire wavelength,” Opt. Lett. 41, 1777–1780 (2016).
[Crossref]

C. L. Korpa, G. Tóth, and J. Hebling, “Interplay of diffraction and nonlinear effects in the propagation of ultrashort pulses,” J. Phys. B 49, 035401 (2016).
[Crossref]

2015 (2)

K. Dolgaleva, D. Materikina, R. Boyd, and S. Kozlov, “Prediction of an extremely large nonlinear refractive index for crystals at terahertz frequencies,” Phys. Rev. A 92, 023809 (2015).
[Crossref]

J. Lu, H. Y. Hwang, X. Li, S.-H. Lee, O.-P. Kwon, and K. A. Nelson, “Tunable multi-cycle THz generation in organic crystal HMQ-TMS,” Opt. Express 23, 22723–22729 (2015).
[Crossref]

2014 (2)

C. Vicario, B. Monoszlai, and C. P. Hauri, “GV/m single-cycle terahertz fields from a laser-driven large-size partitioned organic crystal,” Phys. Rev. Lett. 112, 213901 (2014).
[Crossref]

L. Pálfalvi, J. A. Fülöp, G. Tóth, and J. Hebling, “Evanescent-wave proton postaccelerator driven by intense THz pulse,” Phys. Rev. Spec. Top. Accel. Beams 17, 031301 (2014).
[Crossref]

2013 (2)

2012 (2)

A. A. Drozdov, S. A. Kozlov, A. A. Sukhorukov, and Y. S. Kivshar, “Self-phase modulation and frequency generation with few-cycle optical pulses in nonlinear dispersive media,” Phys. Rev. A 86, 053822 (2012).
[Crossref]

I. Katayama, H. Aoki, J. Takeda, H. Shimosato, M. Ashida, R. Kinjo, I. Kawayama, M. Tonouchi, M. Nagai, and K. Tanaka, “Ferroelectric soft mode in a srtio3 thin film impulsively driven to the anharmonic regime using intense picosecond terahertz pulses,” Phys. Rev. Lett. 108, 097401 (2012).
[Crossref]

2010 (1)

2009 (2)

T. Qi, Y.-H. Shin, K.-L. Yeh, K. A. Nelson, and A. M. Rappe, “Collective coherent control: synchronization of polarization in ferroelectric pbtio3 by shaped THz fields,” Phys. Rev. Lett. 102, 247603 (2009).
[Crossref]

V. P. Kandidov, S. A. Shlenov, and O. G. Kosareva, “Filamentation of high-power femtosecond laser radiation,” Quantum Electron. 39, 205 (2009).
[Crossref]

2008 (1)

2005 (1)

A. N. Berkovsky, S. A. Kozlov, and Y. A. Shpolyanskiy, “Self-focusing of few-cycle light pulses in dielectric media,” Phys. Rev. A 72, 043821 (2005).
[Crossref]

2001 (1)

S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propagation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. 87, 213902 (2001).
[Crossref]

2000 (2)

G. Fibich and A. L. Gaeta, “Critical power for self-focusing in bulk media and in hollow waveguides,” Opt. Lett. 25, 335–337 (2000).
[Crossref]

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[Crossref]

1998 (1)

1997 (2)

S. A. Kozlov and S. V. Sazonov, “Nonlinear propagation of optical pulses of a few oscillations duration in dielectric media,” JETP 84, 221–228 (1997).
[Crossref]

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[Crossref]

1994 (2)

1992 (2)

1964 (1)

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479 (1964).
[Crossref]

1962 (1)

G. Askaryan, “Effects of the gradient of a strong electromagnetic beam on electrons and atoms,” Sov. Phys. JETP-USSR 15, 1088–1090 (1962).

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics, 5th ed. (Academic, 2013).

Ahr, F.

S. W. Jolly, N. H. Matlis, F. Ahr, V. Leroux, T. Eichner, A.-L. Calendron, H. Ishizuki, T. Taira, F. X. Kärtner, and A. R. Maier, “Spectral phase control of interfering chirped pulses for high-energy narrowband terahertz generation,” Nat. Commun. 10, 2591 (2019).
[Crossref]

Aoki, H.

I. Katayama, H. Aoki, J. Takeda, H. Shimosato, M. Ashida, R. Kinjo, I. Kawayama, M. Tonouchi, M. Nagai, and K. Tanaka, “Ferroelectric soft mode in a srtio3 thin film impulsively driven to the anharmonic regime using intense picosecond terahertz pulses,” Phys. Rev. Lett. 108, 097401 (2012).
[Crossref]

Ashida, M.

I. Katayama, H. Aoki, J. Takeda, H. Shimosato, M. Ashida, R. Kinjo, I. Kawayama, M. Tonouchi, M. Nagai, and K. Tanaka, “Ferroelectric soft mode in a srtio3 thin film impulsively driven to the anharmonic regime using intense picosecond terahertz pulses,” Phys. Rev. Lett. 108, 097401 (2012).
[Crossref]

Askaryan, G.

G. Askaryan, “Effects of the gradient of a strong electromagnetic beam on electrons and atoms,” Sov. Phys. JETP-USSR 15, 1088–1090 (1962).

Bergé, L.

S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propagation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. 87, 213902 (2001).
[Crossref]

Berkovskii, A. N.

Berkovsky, A. N.

A. N. Berkovsky, S. A. Kozlov, and Y. A. Shpolyanskiy, “Self-focusing of few-cycle light pulses in dielectric media,” Phys. Rev. A 72, 043821 (2005).
[Crossref]

Bespalov, V. G.

A. N. Tcypkin, S. E. Putilin, M. C. Kulya, M. V. Melnik, A. A. Drozdov, V. G. Bespalov, X.-C. Zhang, R. W. Boyd, and S. A. Kozlov, “Experimental estimate of the nonlinear refractive index of crystalline ZnSe in the terahertz spectral range,” Bull. Russ. Acad. Sci. Phys. 82(12), 1547–1549 (2018).
[Crossref]

Boyd, R.

K. Dolgaleva, D. Materikina, R. Boyd, and S. Kozlov, “Prediction of an extremely large nonlinear refractive index for crystals at terahertz frequencies,” Phys. Rev. A 92, 023809 (2015).
[Crossref]

Boyd, R. W.

A. N. Tcypkin, S. E. Putilin, M. C. Kulya, M. V. Melnik, A. A. Drozdov, V. G. Bespalov, X.-C. Zhang, R. W. Boyd, and S. A. Kozlov, “Experimental estimate of the nonlinear refractive index of crystalline ZnSe in the terahertz spectral range,” Bull. Russ. Acad. Sci. Phys. 82(12), 1547–1549 (2018).
[Crossref]

R. W. Boyd, Nonlinear Optics (Elsevier, 2003).

R. W. Boyd, S. Lukishova, and Y. Shen, Self-focusing: Past and Present (Springer, 2009).

Brabec, T.

T. Brabec and F. Krausz, “Intense few-cycle laser fields: frontiers of nonlinear optics,” Rev. Mod. Phys. 72, 545–591 (2000).
[Crossref]

T. Brabec and F. Krausz, “Nonlinear optical pulse propagation in the single-cycle regime,” Phys. Rev. Lett. 78, 3282–3285 (1997).
[Crossref]

Brahadeeswaran, S.

Calendron, A.-L.

S. W. Jolly, N. H. Matlis, F. Ahr, V. Leroux, T. Eichner, A.-L. Calendron, H. Ishizuki, T. Taira, F. X. Kärtner, and A. R. Maier, “Spectral phase control of interfering chirped pulses for high-energy narrowband terahertz generation,” Nat. Commun. 10, 2591 (2019).
[Crossref]

Chernev, P.

Chiao, R. Y.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479 (1964).
[Crossref]

Corkum, P. B.

Couairon, A.

S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propagation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. 87, 213902 (2001).
[Crossref]

Dolgaleva, K.

K. Dolgaleva, D. Materikina, R. Boyd, and S. Kozlov, “Prediction of an extremely large nonlinear refractive index for crystals at terahertz frequencies,” Phys. Rev. A 92, 023809 (2015).
[Crossref]

Dörner, R.

Drozdov, A. A.

A. N. Tcypkin, S. E. Putilin, M. C. Kulya, M. V. Melnik, A. A. Drozdov, V. G. Bespalov, X.-C. Zhang, R. W. Boyd, and S. A. Kozlov, “Experimental estimate of the nonlinear refractive index of crystalline ZnSe in the terahertz spectral range,” Bull. Russ. Acad. Sci. Phys. 82(12), 1547–1549 (2018).
[Crossref]

A. A. Drozdov, S. A. Kozlov, A. A. Sukhorukov, and Y. S. Kivshar, “Self-phase modulation and frequency generation with few-cycle optical pulses in nonlinear dispersive media,” Phys. Rev. A 86, 053822 (2012).
[Crossref]

Eichner, T.

S. W. Jolly, N. H. Matlis, F. Ahr, V. Leroux, T. Eichner, A.-L. Calendron, H. Ishizuki, T. Taira, F. X. Kärtner, and A. R. Maier, “Spectral phase control of interfering chirped pulses for high-energy narrowband terahertz generation,” Nat. Commun. 10, 2591 (2019).
[Crossref]

Fibich, G.

Franco, M.

S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propagation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. 87, 213902 (2001).
[Crossref]

Fülöp, J.

P. Nugraha, G. Krizsán, G. Polónyi, M. Mechler, J. Hebling, G. Tóth, and J. Fülöp, “Efficient semiconductor multicycle terahertz pulse source,” J. Phys. B 51, 094007 (2018).
[Crossref]

Fülöp, J. A.

L. Pálfalvi, J. A. Fülöp, G. Tóth, and J. Hebling, “Evanescent-wave proton postaccelerator driven by intense THz pulse,” Phys. Rev. Spec. Top. Accel. Beams 17, 031301 (2014).
[Crossref]

Gaeta, A.

Gaeta, A. L.

Garmire, E.

R. Y. Chiao, E. Garmire, and C. H. Townes, “Self-trapping of optical beams,” Phys. Rev. Lett. 13, 479 (1964).
[Crossref]

Glover, R. D.

Hauri, C. P.

M. Shalaby, C. Vicario, K. Thirupugalmani, S. Brahadeeswaran, and C. P. Hauri, “Intense THz source based on BNA organic crystal pumped at Ti:sapphire wavelength,” Opt. Lett. 41, 1777–1780 (2016).
[Crossref]

C. Vicario, B. Monoszlai, and C. P. Hauri, “GV/m single-cycle terahertz fields from a laser-driven large-size partitioned organic crystal,” Phys. Rev. Lett. 112, 213901 (2014).
[Crossref]

Hebling, J.

P. Nugraha, G. Krizsán, G. Polónyi, M. Mechler, J. Hebling, G. Tóth, and J. Fülöp, “Efficient semiconductor multicycle terahertz pulse source,” J. Phys. B 51, 094007 (2018).
[Crossref]

C. L. Korpa, G. Tóth, and J. Hebling, “Mapping the lattice-vibration potential using terahertz pulses,” J. Phys. B 51, 035403 (2018).
[Crossref]

C. L. Korpa, G. Tóth, and J. Hebling, “Interplay of diffraction and nonlinear effects in the propagation of ultrashort pulses,” J. Phys. B 49, 035401 (2016).
[Crossref]

L. Pálfalvi, J. A. Fülöp, G. Tóth, and J. Hebling, “Evanescent-wave proton postaccelerator driven by intense THz pulse,” Phys. Rev. Spec. Top. Accel. Beams 17, 031301 (2014).
[Crossref]

J. Hebling, M. C. Hoffmann, K.-L. Yeh, G. Tóth, and K. A. Nelson, “Nonlinear lattice response observed through terahertz SPM,” in Ultrafast Phenomena XVI, P. Corkum, S. Silvestri, K. A. Nelson, E. Riedle, and R. W. Schoenlein, eds., Vol. 92 of Springer Series in Chemical Physics (Springer-Verlag, 2009), pp. 651–653.

Hoffmann, M. C.

J. Hebling, M. C. Hoffmann, K.-L. Yeh, G. Tóth, and K. A. Nelson, “Nonlinear lattice response observed through terahertz SPM,” in Ultrafast Phenomena XVI, P. Corkum, S. Silvestri, K. A. Nelson, E. Riedle, and R. W. Schoenlein, eds., Vol. 92 of Springer Series in Chemical Physics (Springer-Verlag, 2009), pp. 651–653.

Hwang, H. Y.

Ishizuki, H.

S. W. Jolly, N. H. Matlis, F. Ahr, V. Leroux, T. Eichner, A.-L. Calendron, H. Ishizuki, T. Taira, F. X. Kärtner, and A. R. Maier, “Spectral phase control of interfering chirped pulses for high-energy narrowband terahertz generation,” Nat. Commun. 10, 2591 (2019).
[Crossref]

Jahnke, T.

Jaiswal, S.

T. Seifert, S. Jaiswal, M. Sajadi, G. Jakob, S. Winnerl, M. Wolf, M. Klaui, and T. Kampfrath, “Ultrabroadband single-cycle terahertz pulses with peak fields of 300  kv  cm-1 from a metallic spintronic emitter,” Appl. Phys. Lett. 110, 252402 (2017).
[Crossref]

Jakob, G.

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S. W. Jolly, N. H. Matlis, F. Ahr, V. Leroux, T. Eichner, A.-L. Calendron, H. Ishizuki, T. Taira, F. X. Kärtner, and A. R. Maier, “Spectral phase control of interfering chirped pulses for high-energy narrowband terahertz generation,” Nat. Commun. 10, 2591 (2019).
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T. Seifert, S. Jaiswal, M. Sajadi, G. Jakob, S. Winnerl, M. Wolf, M. Klaui, and T. Kampfrath, “Ultrabroadband single-cycle terahertz pulses with peak fields of 300  kv  cm-1 from a metallic spintronic emitter,” Appl. Phys. Lett. 110, 252402 (2017).
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T. Seifert, S. Jaiswal, M. Sajadi, G. Jakob, S. Winnerl, M. Wolf, M. Klaui, and T. Kampfrath, “Ultrabroadband single-cycle terahertz pulses with peak fields of 300  kv  cm-1 from a metallic spintronic emitter,” Appl. Phys. Lett. 110, 252402 (2017).
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C. L. Korpa, G. Tóth, and J. Hebling, “Mapping the lattice-vibration potential using terahertz pulses,” J. Phys. B 51, 035403 (2018).
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C. L. Korpa, G. Tóth, and J. Hebling, “Interplay of diffraction and nonlinear effects in the propagation of ultrashort pulses,” J. Phys. B 49, 035401 (2016).
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M. Melnik, I. Vorontsova, S. Putilin, A. Tcypkin, and S. Kozlov, “Methodical inaccuracy of the z-scan method for few-cycle terahertz pulses,” Sci. Rep. 9, 9146 (2019).
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K. Dolgaleva, D. Materikina, R. Boyd, and S. Kozlov, “Prediction of an extremely large nonlinear refractive index for crystals at terahertz frequencies,” Phys. Rev. A 92, 023809 (2015).
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Kozlov, S. A.

A. N. Tcypkin, M. V. Melnik, M. O. Zhukova, I. O. Vorontsova, S. E. Putilin, S. A. Kozlov, and X.-C. Zhang, “High Kerr nonlinearity of water in THz spectral range,” Opt. Express 27, 10419–10425 (2019).
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A. N. Tcypkin, S. E. Putilin, M. C. Kulya, M. V. Melnik, A. A. Drozdov, V. G. Bespalov, X.-C. Zhang, R. W. Boyd, and S. A. Kozlov, “Experimental estimate of the nonlinear refractive index of crystalline ZnSe in the terahertz spectral range,” Bull. Russ. Acad. Sci. Phys. 82(12), 1547–1549 (2018).
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A. A. Drozdov, S. A. Kozlov, A. A. Sukhorukov, and Y. S. Kivshar, “Self-phase modulation and frequency generation with few-cycle optical pulses in nonlinear dispersive media,” Phys. Rev. A 86, 053822 (2012).
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A. N. Berkovskii, S. A. Kozlov, and Y. A. Shpolyanskii, “Reducing the self-focusing efficiency of a femtosecond pulse in a transparent medium with dispersion when the number of light vibrations in it is decreased,” J. Opt. Technol. 75, 631–635 (2008).
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A. N. Berkovsky, S. A. Kozlov, and Y. A. Shpolyanskiy, “Self-focusing of few-cycle light pulses in dielectric media,” Phys. Rev. A 72, 043821 (2005).
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S. A. Kozlov and S. V. Sazonov, “Nonlinear propagation of optical pulses of a few oscillations duration in dielectric media,” JETP 84, 221–228 (1997).
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A. N. Tcypkin, S. E. Putilin, M. C. Kulya, M. V. Melnik, A. A. Drozdov, V. G. Bespalov, X.-C. Zhang, R. W. Boyd, and S. A. Kozlov, “Experimental estimate of the nonlinear refractive index of crystalline ZnSe in the terahertz spectral range,” Bull. Russ. Acad. Sci. Phys. 82(12), 1547–1549 (2018).
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Kwon, O.-P.

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S. W. Jolly, N. H. Matlis, F. Ahr, V. Leroux, T. Eichner, A.-L. Calendron, H. Ishizuki, T. Taira, F. X. Kärtner, and A. R. Maier, “Spectral phase control of interfering chirped pulses for high-energy narrowband terahertz generation,” Nat. Commun. 10, 2591 (2019).
[Crossref]

Materikina, D.

K. Dolgaleva, D. Materikina, R. Boyd, and S. Kozlov, “Prediction of an extremely large nonlinear refractive index for crystals at terahertz frequencies,” Phys. Rev. A 92, 023809 (2015).
[Crossref]

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S. W. Jolly, N. H. Matlis, F. Ahr, V. Leroux, T. Eichner, A.-L. Calendron, H. Ishizuki, T. Taira, F. X. Kärtner, and A. R. Maier, “Spectral phase control of interfering chirped pulses for high-energy narrowband terahertz generation,” Nat. Commun. 10, 2591 (2019).
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P. Nugraha, G. Krizsán, G. Polónyi, M. Mechler, J. Hebling, G. Tóth, and J. Fülöp, “Efficient semiconductor multicycle terahertz pulse source,” J. Phys. B 51, 094007 (2018).
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M. Melnik, I. Vorontsova, S. Putilin, A. Tcypkin, and S. Kozlov, “Methodical inaccuracy of the z-scan method for few-cycle terahertz pulses,” Sci. Rep. 9, 9146 (2019).
[Crossref]

Melnik, M. V.

A. N. Tcypkin, M. V. Melnik, M. O. Zhukova, I. O. Vorontsova, S. E. Putilin, S. A. Kozlov, and X.-C. Zhang, “High Kerr nonlinearity of water in THz spectral range,” Opt. Express 27, 10419–10425 (2019).
[Crossref]

A. N. Tcypkin, S. E. Putilin, M. C. Kulya, M. V. Melnik, A. A. Drozdov, V. G. Bespalov, X.-C. Zhang, R. W. Boyd, and S. A. Kozlov, “Experimental estimate of the nonlinear refractive index of crystalline ZnSe in the terahertz spectral range,” Bull. Russ. Acad. Sci. Phys. 82(12), 1547–1549 (2018).
[Crossref]

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Monoszlai, B.

C. Vicario, B. Monoszlai, and C. P. Hauri, “GV/m single-cycle terahertz fields from a laser-driven large-size partitioned organic crystal,” Phys. Rev. Lett. 112, 213901 (2014).
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S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propagation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. 87, 213902 (2001).
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I. Katayama, H. Aoki, J. Takeda, H. Shimosato, M. Ashida, R. Kinjo, I. Kawayama, M. Tonouchi, M. Nagai, and K. Tanaka, “Ferroelectric soft mode in a srtio3 thin film impulsively driven to the anharmonic regime using intense picosecond terahertz pulses,” Phys. Rev. Lett. 108, 097401 (2012).
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J. Lu, H. Y. Hwang, X. Li, S.-H. Lee, O.-P. Kwon, and K. A. Nelson, “Tunable multi-cycle THz generation in organic crystal HMQ-TMS,” Opt. Express 23, 22723–22729 (2015).
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T. Qi, Y.-H. Shin, K.-L. Yeh, K. A. Nelson, and A. M. Rappe, “Collective coherent control: synchronization of polarization in ferroelectric pbtio3 by shaped THz fields,” Phys. Rev. Lett. 102, 247603 (2009).
[Crossref]

J. Hebling, M. C. Hoffmann, K.-L. Yeh, G. Tóth, and K. A. Nelson, “Nonlinear lattice response observed through terahertz SPM,” in Ultrafast Phenomena XVI, P. Corkum, S. Silvestri, K. A. Nelson, E. Riedle, and R. W. Schoenlein, eds., Vol. 92 of Springer Series in Chemical Physics (Springer-Verlag, 2009), pp. 651–653.

Newell, A.

Nugraha, P.

P. Nugraha, G. Krizsán, G. Polónyi, M. Mechler, J. Hebling, G. Tóth, and J. Fülöp, “Efficient semiconductor multicycle terahertz pulse source,” J. Phys. B 51, 094007 (2018).
[Crossref]

Pálfalvi, L.

L. Pálfalvi, J. A. Fülöp, G. Tóth, and J. Hebling, “Evanescent-wave proton postaccelerator driven by intense THz pulse,” Phys. Rev. Spec. Top. Accel. Beams 17, 031301 (2014).
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Petrov, V.

Polónyi, G.

P. Nugraha, G. Krizsán, G. Polónyi, M. Mechler, J. Hebling, G. Tóth, and J. Fülöp, “Efficient semiconductor multicycle terahertz pulse source,” J. Phys. B 51, 094007 (2018).
[Crossref]

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P. Polynkin and M. Kolesik, “Critical power for self-focusing in the case of ultrashort laser pulses,” Phys. Rev. A 87, 053829 (2013).
[Crossref]

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S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propagation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. 87, 213902 (2001).
[Crossref]

Putilin, S.

M. Melnik, I. Vorontsova, S. Putilin, A. Tcypkin, and S. Kozlov, “Methodical inaccuracy of the z-scan method for few-cycle terahertz pulses,” Sci. Rep. 9, 9146 (2019).
[Crossref]

Putilin, S. E.

A. N. Tcypkin, M. V. Melnik, M. O. Zhukova, I. O. Vorontsova, S. E. Putilin, S. A. Kozlov, and X.-C. Zhang, “High Kerr nonlinearity of water in THz spectral range,” Opt. Express 27, 10419–10425 (2019).
[Crossref]

A. N. Tcypkin, S. E. Putilin, M. C. Kulya, M. V. Melnik, A. A. Drozdov, V. G. Bespalov, X.-C. Zhang, R. W. Boyd, and S. A. Kozlov, “Experimental estimate of the nonlinear refractive index of crystalline ZnSe in the terahertz spectral range,” Bull. Russ. Acad. Sci. Phys. 82(12), 1547–1549 (2018).
[Crossref]

Qi, T.

T. Qi, Y.-H. Shin, K.-L. Yeh, K. A. Nelson, and A. M. Rappe, “Collective coherent control: synchronization of polarization in ferroelectric pbtio3 by shaped THz fields,” Phys. Rev. Lett. 102, 247603 (2009).
[Crossref]

Ranka, J.

Rappe, A. M.

T. Qi, Y.-H. Shin, K.-L. Yeh, K. A. Nelson, and A. M. Rappe, “Collective coherent control: synchronization of polarization in ferroelectric pbtio3 by shaped THz fields,” Phys. Rev. Lett. 102, 247603 (2009).
[Crossref]

Richter, M.

Roskos, H. G.

Rothenberg, J. E.

Sajadi, M.

T. Seifert, S. Jaiswal, M. Sajadi, G. Jakob, S. Winnerl, M. Wolf, M. Klaui, and T. Kampfrath, “Ultrabroadband single-cycle terahertz pulses with peak fields of 300  kv  cm-1 from a metallic spintronic emitter,” Appl. Phys. Lett. 110, 252402 (2017).
[Crossref]

Samartsev, V. V.

S. A. Kozlov and V. V. Samartsev, Fundamentals of Femtosecond Optics (Woodhead, 2013).

Sang, R. T.

Sazonov, S. V.

S. A. Kozlov and S. V. Sazonov, “Nonlinear propagation of optical pulses of a few oscillations duration in dielectric media,” JETP 84, 221–228 (1997).
[Crossref]

Schmidt-Böcking, H.

Schöffler, M.

Seifert, T.

T. Seifert, S. Jaiswal, M. Sajadi, G. Jakob, S. Winnerl, M. Wolf, M. Klaui, and T. Kampfrath, “Ultrabroadband single-cycle terahertz pulses with peak fields of 300  kv  cm-1 from a metallic spintronic emitter,” Appl. Phys. Lett. 110, 252402 (2017).
[Crossref]

Shalaby, M.

Shen, Y.

R. W. Boyd, S. Lukishova, and Y. Shen, Self-focusing: Past and Present (Springer, 2009).

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I. Katayama, H. Aoki, J. Takeda, H. Shimosato, M. Ashida, R. Kinjo, I. Kawayama, M. Tonouchi, M. Nagai, and K. Tanaka, “Ferroelectric soft mode in a srtio3 thin film impulsively driven to the anharmonic regime using intense picosecond terahertz pulses,” Phys. Rev. Lett. 108, 097401 (2012).
[Crossref]

Shin, Y.-H.

T. Qi, Y.-H. Shin, K.-L. Yeh, K. A. Nelson, and A. M. Rappe, “Collective coherent control: synchronization of polarization in ferroelectric pbtio3 by shaped THz fields,” Phys. Rev. Lett. 102, 247603 (2009).
[Crossref]

Shlenov, S. A.

V. P. Kandidov, S. A. Shlenov, and O. G. Kosareva, “Filamentation of high-power femtosecond laser radiation,” Quantum Electron. 39, 205 (2009).
[Crossref]

Shpolyanskii, Y. A.

Shpolyanskiy, Y. A.

A. N. Berkovsky, S. A. Kozlov, and Y. A. Shpolyanskiy, “Self-focusing of few-cycle light pulses in dielectric media,” Phys. Rev. A 72, 043821 (2005).
[Crossref]

Strickland, D.

Sudrie, L.

S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propagation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. 87, 213902 (2001).
[Crossref]

Sukhorukov, A. A.

A. A. Drozdov, S. A. Kozlov, A. A. Sukhorukov, and Y. S. Kivshar, “Self-phase modulation and frequency generation with few-cycle optical pulses in nonlinear dispersive media,” Phys. Rev. A 86, 053822 (2012).
[Crossref]

Taira, T.

S. W. Jolly, N. H. Matlis, F. Ahr, V. Leroux, T. Eichner, A.-L. Calendron, H. Ishizuki, T. Taira, F. X. Kärtner, and A. R. Maier, “Spectral phase control of interfering chirped pulses for high-energy narrowband terahertz generation,” Nat. Commun. 10, 2591 (2019).
[Crossref]

Takeda, J.

I. Katayama, H. Aoki, J. Takeda, H. Shimosato, M. Ashida, R. Kinjo, I. Kawayama, M. Tonouchi, M. Nagai, and K. Tanaka, “Ferroelectric soft mode in a srtio3 thin film impulsively driven to the anharmonic regime using intense picosecond terahertz pulses,” Phys. Rev. Lett. 108, 097401 (2012).
[Crossref]

Tanaka, K.

I. Katayama, H. Aoki, J. Takeda, H. Shimosato, M. Ashida, R. Kinjo, I. Kawayama, M. Tonouchi, M. Nagai, and K. Tanaka, “Ferroelectric soft mode in a srtio3 thin film impulsively driven to the anharmonic regime using intense picosecond terahertz pulses,” Phys. Rev. Lett. 108, 097401 (2012).
[Crossref]

Tcypkin, A.

M. Melnik, I. Vorontsova, S. Putilin, A. Tcypkin, and S. Kozlov, “Methodical inaccuracy of the z-scan method for few-cycle terahertz pulses,” Sci. Rep. 9, 9146 (2019).
[Crossref]

Tcypkin, A. N.

A. N. Tcypkin, M. V. Melnik, M. O. Zhukova, I. O. Vorontsova, S. E. Putilin, S. A. Kozlov, and X.-C. Zhang, “High Kerr nonlinearity of water in THz spectral range,” Opt. Express 27, 10419–10425 (2019).
[Crossref]

A. N. Tcypkin, S. E. Putilin, M. C. Kulya, M. V. Melnik, A. A. Drozdov, V. G. Bespalov, X.-C. Zhang, R. W. Boyd, and S. A. Kozlov, “Experimental estimate of the nonlinear refractive index of crystalline ZnSe in the terahertz spectral range,” Bull. Russ. Acad. Sci. Phys. 82(12), 1547–1549 (2018).
[Crossref]

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Thomson, M. D.

Tonouchi, M.

I. Katayama, H. Aoki, J. Takeda, H. Shimosato, M. Ashida, R. Kinjo, I. Kawayama, M. Tonouchi, M. Nagai, and K. Tanaka, “Ferroelectric soft mode in a srtio3 thin film impulsively driven to the anharmonic regime using intense picosecond terahertz pulses,” Phys. Rev. Lett. 108, 097401 (2012).
[Crossref]

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C. L. Korpa, G. Tóth, and J. Hebling, “Mapping the lattice-vibration potential using terahertz pulses,” J. Phys. B 51, 035403 (2018).
[Crossref]

P. Nugraha, G. Krizsán, G. Polónyi, M. Mechler, J. Hebling, G. Tóth, and J. Fülöp, “Efficient semiconductor multicycle terahertz pulse source,” J. Phys. B 51, 094007 (2018).
[Crossref]

C. L. Korpa, G. Tóth, and J. Hebling, “Interplay of diffraction and nonlinear effects in the propagation of ultrashort pulses,” J. Phys. B 49, 035401 (2016).
[Crossref]

L. Pálfalvi, J. A. Fülöp, G. Tóth, and J. Hebling, “Evanescent-wave proton postaccelerator driven by intense THz pulse,” Phys. Rev. Spec. Top. Accel. Beams 17, 031301 (2014).
[Crossref]

J. Hebling, M. C. Hoffmann, K.-L. Yeh, G. Tóth, and K. A. Nelson, “Nonlinear lattice response observed through terahertz SPM,” in Ultrafast Phenomena XVI, P. Corkum, S. Silvestri, K. A. Nelson, E. Riedle, and R. W. Schoenlein, eds., Vol. 92 of Springer Series in Chemical Physics (Springer-Verlag, 2009), pp. 651–653.

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S. Tzortzakis, L. Sudrie, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and L. Bergé, “Self-guided propagation of ultrashort IR laser pulses in fused silica,” Phys. Rev. Lett. 87, 213902 (2001).
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M. Shalaby, C. Vicario, K. Thirupugalmani, S. Brahadeeswaran, and C. P. Hauri, “Intense THz source based on BNA organic crystal pumped at Ti:sapphire wavelength,” Opt. Lett. 41, 1777–1780 (2016).
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C. Vicario, B. Monoszlai, and C. P. Hauri, “GV/m single-cycle terahertz fields from a laser-driven large-size partitioned organic crystal,” Phys. Rev. Lett. 112, 213901 (2014).
[Crossref]

Vorontsova, I.

M. Melnik, I. Vorontsova, S. Putilin, A. Tcypkin, and S. Kozlov, “Methodical inaccuracy of the z-scan method for few-cycle terahertz pulses,” Sci. Rep. 9, 9146 (2019).
[Crossref]

Vorontsova, I. O.

Vredenborg, A.

Wallace, W. C.

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T. Seifert, S. Jaiswal, M. Sajadi, G. Jakob, S. Winnerl, M. Wolf, M. Klaui, and T. Kampfrath, “Ultrabroadband single-cycle terahertz pulses with peak fields of 300  kv  cm-1 from a metallic spintronic emitter,” Appl. Phys. Lett. 110, 252402 (2017).
[Crossref]

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T. Seifert, S. Jaiswal, M. Sajadi, G. Jakob, S. Winnerl, M. Wolf, M. Klaui, and T. Kampfrath, “Ultrabroadband single-cycle terahertz pulses with peak fields of 300  kv  cm-1 from a metallic spintronic emitter,” Appl. Phys. Lett. 110, 252402 (2017).
[Crossref]

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Wu, J.

Yeh, K.-L.

T. Qi, Y.-H. Shin, K.-L. Yeh, K. A. Nelson, and A. M. Rappe, “Collective coherent control: synchronization of polarization in ferroelectric pbtio3 by shaped THz fields,” Phys. Rev. Lett. 102, 247603 (2009).
[Crossref]

J. Hebling, M. C. Hoffmann, K.-L. Yeh, G. Tóth, and K. A. Nelson, “Nonlinear lattice response observed through terahertz SPM,” in Ultrafast Phenomena XVI, P. Corkum, S. Silvestri, K. A. Nelson, E. Riedle, and R. W. Schoenlein, eds., Vol. 92 of Springer Series in Chemical Physics (Springer-Verlag, 2009), pp. 651–653.

Zhang, X.-C.

A. N. Tcypkin, M. V. Melnik, M. O. Zhukova, I. O. Vorontsova, S. E. Putilin, S. A. Kozlov, and X.-C. Zhang, “High Kerr nonlinearity of water in THz spectral range,” Opt. Express 27, 10419–10425 (2019).
[Crossref]

A. N. Tcypkin, S. E. Putilin, M. C. Kulya, M. V. Melnik, A. A. Drozdov, V. G. Bespalov, X.-C. Zhang, R. W. Boyd, and S. A. Kozlov, “Experimental estimate of the nonlinear refractive index of crystalline ZnSe in the terahertz spectral range,” Bull. Russ. Acad. Sci. Phys. 82(12), 1547–1549 (2018).
[Crossref]

Zhukova, M. O.

Appl. Phys. Lett. (1)

T. Seifert, S. Jaiswal, M. Sajadi, G. Jakob, S. Winnerl, M. Wolf, M. Klaui, and T. Kampfrath, “Ultrabroadband single-cycle terahertz pulses with peak fields of 300  kv  cm-1 from a metallic spintronic emitter,” Appl. Phys. Lett. 110, 252402 (2017).
[Crossref]

Bull. Russ. Acad. Sci. Phys. (1)

A. N. Tcypkin, S. E. Putilin, M. C. Kulya, M. V. Melnik, A. A. Drozdov, V. G. Bespalov, X.-C. Zhang, R. W. Boyd, and S. A. Kozlov, “Experimental estimate of the nonlinear refractive index of crystalline ZnSe in the terahertz spectral range,” Bull. Russ. Acad. Sci. Phys. 82(12), 1547–1549 (2018).
[Crossref]

J. Opt. Soc. Am. B (1)

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Figures (5)

Fig. 1.
Fig. 1. Left: spatiotemporal evolution of the electric field amplitude $ E $ of a near-IR pulse propagating through fused silica. Right: modulus $ |G| $ of the frequency spectrum on the beam axis for the same propagation distances $ \tilde z $ , with the time-varying field $ |E| $ in the insets. The parameters of the input pulse are: $ {\lambda _0} = 800\,\,{\text{nm}} $ , $ {r_0} = 30{\lambda _0} $ , $ {\tau _0} = 27\,\,{\text{fs}} $ (i.e., $ N = 20 $ ), $ I = 5 \times {10^{11}}\,\,{\text{W}}/{\text{c}}{{\text{m}}^2} $ . Here, $ \tau = t - z/{V_g} $ is the retarded time, $ \tilde r = r/{r_0} $ , $ \tilde \tau = \tau /{\tau _0} $ , $ \tilde \omega = \omega /{\omega _0} $ . We see that self-focusing dynamics is not appreciably influenced by pulse duration effects under the conditions reported here. The spatiotemporal evolution shown here is consistent with reported results [5,6,9,28].
Fig. 2.
Fig. 2. Left: spatiotemporal evolution of the electric field amplitude $ E $ of a near-IR few-cycle pulse propagating through fused silica. Right: modulus $ |G| $ of the frequency spectrum on the beam axis for the same propagation distances $ \tilde z $ . The parameters of the input pulse are: $ {\lambda _0} = 800\,\,{\text{nm}} $ , $ {r_0} = 30{\lambda _0} $ , $ {\tau _0} = 8\,\,{\text{fs}} $ (i.e., $ N = 6 $ ), and $ I = 5 \times {10^{11}}\,\,{\text{W}}/{\text{c}}{{\text{m}}^2} $ , corresponding to $ P = 4{P_{\text{cr}}} $ . Instead of transverse and longitudinal compression of the pulse as observed in Figs. 1(b) and 1(c), here, we observe significant temporal broadening due to the strong GVD, but no transverse compression (i.e., no self-focusing). We see that, for this few-cycle optical pulse, the strong normal GVD is able to inhibit self-focusing even though $ P = 4{P_{\text{cr}}} $ .
Fig. 3.
Fig. 3. Left: spatiotemporal evolution of the electric field amplitude $ E $ of a single-cycle THz pulse propagating in a $ {\text{MgO}}:{\text{LiNb}}{{\text{O}}_3} $ crystal. Right: modulus $ |G| $ of the frequency spectrum on the beam axis for the same propagation distances $ \tilde z $ . The parameters of the input pulse are: $ {\lambda _0} = 300\,\,\unicode{x00B5} {\text{m}} $ , $ {r_0} = 5{\lambda _0} $ , $ {\tau _0} = 0.3\,\,{\text{ps}} $ (i.e., $ N = 0.6 $ ), and $ I = 3 \times {10^8}\,\,{\text{W}}/{\text{c}}{{\text{m}}^2} $ corresponding to $ P = 4{P_{\text{cr}}} $ . For the single-cycle THz pulse considered here, we again see that the strong dispersive spreading of the pulse overcomes self-focusing even for $ P = 4{P_{\text{cr}}} $ , and dominates the evolution of the pulse as it propagates through the medium.
Fig. 4.
Fig. 4. Left: spatiotemporal evolution of the electric field amplitude $ E $ of a very intense, single-cycle THz pulse propagating in a $ {\text{MgO:LiNb}}{{\text{O}}_3} $ crystal. Right: modulus $ |G| $ of the frequency spectrum on the beam axis for the same propagation distances $ \tilde z $ . The parameters of the input pulse are: $ {\lambda _0} = 300\,\,{\unicode{x00B5}} {\text{m}} $ , $ {r_0} = 5{\lambda _0} $ , $ {\tau _0} = 0.3\,\,{\text{ps}} $ , $ I = 1.48 \times {10^{10}}\,\,{\text{W}}/{\text{c}}{{\text{m}}^2} $ , corresponding to $ P = 200{P_{\text{cr}}} $ . The panel (d’) shows the output of propagating the pulse in the left panel of (c) through the entire length of the medium while considering only linear propagation, or equivalently with the nonlinear contributions neglected ( $ {n^\prime _2} = 0 $ ). In contrast with the spatiotemporal evolution of a single-cycle THz pulse for $ P = 4{P_{\text{cr}}} $ shown in Fig. 3, here, we find that transverse compression (self-focusing) does occur. We observe self-focusing of the pulse [left, panels (c) and (d)] followed by dispersive-diffractive spreading of the pulse with further propagation [left, panel (e)].
Fig. 5.
Fig. 5. Left: spatiotemporal evolution of the electric field amplitude $ E $ of the few-cycle THz pulse propagating in a $ {\text{MgO:LiNb}}{{\text{O}}_3} $ crystal. Right: modulus $ |G| $ of the frequency spectrum on the beam axis for the same propagation distances $ \tilde z $ , with the time-varying field $ |E| $ in the insets. The parameters of the input pulse are: $ {\lambda _0} = 300\;\unicode{x00B5} {\text{m}} $ , $ {r_0} = 5{\lambda _0} $ , $ {\tau _0} = 8\;{\text{ps}} $ (i.e., $ N = 16 $ ), $ I = 3 \times {10^8}\;{\text{W}}/{\text{c}}{{\text{m}}^2} $ , and $ {P_0} = 4{P_{{\text{cr}}}} $ , where $ {P_{{\text{cr}}}} = 5.3 \times {10^6}W $ . For the multiple-optical-cycles-long THz pulse shown here, the spatial evolution of the pulse is functionally similar to the evolution of multi-cycle near-IR pulses propagating in a normally dispersive medium, shown in Fig. 1.

Equations (21)

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E ~ z + 1 L w env E ~ t ~ + i L disp1 env 2 E ~ t ~ 2 1 L disp2 env 3 E ~ t ~ 3 i L nl1 env | E ~ | 2 E ~ + 1 L nl2 env t ~ ( | E ~ | 2 E ~ ) = i L diffr env Δ ~ E ~ ,
L nl1 env = L diffr env ,
P cr = R cr λ 0 2 8 π n 0 n 2 ,
L disp1 env < L diffr env .
l 0 D 0 < c ω 0 n ( ω 0 ) β 2 ,
n ( ω ) = N 0 + a c ω 2 ,
l 0 D 0 < 6 N 0 Δ n disp ,
n ( ω ) = N 0 + a c ω 2 b c / ω 2 ,
E ( 0 , r , t ) = E ( 0 , r , t ) sin ( ω 0 t ) ,
E ( 0 , r , t ) = E 0 exp ( r 2 r 0 2 ) exp ( t 2 τ 0 2 ) ,
E ~ ( 0 , r ~ , t ~ ) = exp ( r ~ 2 ) exp ( t ~ 2 ) .
E ~ z + 1 L wave f E ~ t ~ 1 L disp 1 f 3 E ~ t ~ 3 + 1 L disp 2 f t ~ E ~ d t ~ + 1 L nl f E ~ 2 E ~ t ~ = 1 L diffr f Δ ~ t ~ E ~ d t ~ ,
[ L disp 1 f ] 1 = ( ω 0 τ 0 ) 2 4 ( 1 L disp 1 env + ω 0 τ 0 L disp2 env ) ,
[ L disp2 f ] 1 = ( ω 0 τ 0 ) 2 4 ( 1 L disp 1 env 3 ω 0 τ 0 L disp2 env ) .
G ( z , 0 , ω ) = + E ( z , 0 , t ) exp ( i ω t ) d t .
E z + N 0 c E t a 3 E t 3 + b t E d t + g E 2 E t = c 2 N 0 Δ t E d t ,
E ( r , t ) = 1 2 E ( r , t ) i ( k 0 z ω 0 t ) + c.c .
E z + 1 V E t + i β 2 2 2 E t 2 β 3 6 3 E t 3 n = 4 β n i n + 1 n ! n E t n i γ 1 | E | 2 E + γ 2 t ( | E | 2 E ) ( i γ 1 E 3 γ 2 E 2 E t ) exp ( 2 i ( k 0 z ω 0 t ) ) = i 2 k 0 Δ [ ω 0 i t E ( r , t ) exp ( i ω 0 ( t t ) ) d t ] ,
i 2 k 0 Δ [ ω 0 i t E ( r , t ) exp ( i ω 0 ( t t ) ) d t ] = i 2 k 0 Δ ( E ( r , t ) i ω 0 E ( r , t ) t + ( i ω 0 ) 2 2 E ( r , t ) t 2 . . . ) ,
E z + 1 V E t + i β 2 2 2 E t 2 β 3 6 3 E t 3 i γ 1 | E | 2 E + γ 2 t ( | E | 2 E ) = i 2 k 0 Δ E .
ω 0 i t E ( r , z , t ) exp ( i ω 0 ( t t ) ) d t = [ 1 + i ω 0 t ] 1 × E ( r , z , t ) .

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